Edexcel IGCSE 4MA1 HCF, LCM and Prime Factors: exam questions and answers, notes and worked examples

HCF, LCM and Prime Factors notes

Factors, multiples and primes

A prime has exactly two factors, 1 and itself, so 1 is not prime and 2 is the only even prime. Factors come in pairs: $1\times 36$, $2\times 18$, $3\times 12$, $4\times 9$, $6\times 6$, so 36 has 9 factors. The multiples of 6 are 6, 12, 18, 24 and so on.

Product of prime factors

Divide by the smallest prime that works, again and again (a factor tree does the same job). $360=2\times 2\times 2\times 3\times 3\times 5=2^3\times 3^2\times 5$. Give the answer in index form when the question asks for it, and check by multiplying back.

HCF and LCM from prime factors

With $24=2^3\times 3$ and $36=2^2\times 3^2$: HCF $=2^2\times 3=12$ and LCM $=2^3\times 3^2=72$. A Venn diagram of the prime factors shows both at once: the overlap multiplies to the HCF and the whole diagram to the LCM. For two numbers, HCF × LCM = 24 × 36 is a quick check.

HCF and LCM problems

LCM questions ask when things happen together again: buses every 12 and 18 minutes leave together again after 36 minutes. HCF questions ask for the largest equal groups: 24 pens and 36 pencils shared into identical packs with nothing left over makes at most 12 packs.

Squares and cubes from prime factors

A number is a square number when every index in its prime factorisation is even, and a cube number when every index is a multiple of 3. $72=2^3\times 3^2$ is not square, but multiplying by 2 gives $2^4\times 3^2=144={12}^2$.